The right acute angles problem?

Abstract

The Danzer--Gr\"unbaum acute angles problem asks for the largest size of a set of points in ${\mathbb R}^d$ that determines only acute angles. Recently, the problem was essentially solved thanks to the results of the second author and of Gerencs\'er and Harangi: now, the lower and the upper bounds are $2^{d-1}+1$ and $2^d-1$, respectively. The lower-bound construction is surprisingly simple. In this note, we suggest the following variant of the problem, which is one way to "save" the problem. Put $F(\alpha) = \lim_{d\to \infty} f(d,\alpha)^{1/d}$, where $f(d,\alpha)$ is the largest set of points in ${\mathbb R}^d$ with no angle greater than $\alpha$. Then the question is to find $c:= \lim_{\alpha\to \pi/2^-}F(\alpha).$ Although one may expect that $c=2$ in view of the result of Gerencs\'er and Harangi, the best lower bound we could get is $c\ge \sqrt 2$. We also solve a related problem of Erdos and F\"uredi on the "stability" of the acute angles problem and refute another conjecture stated in the same paper.